- 1.
- Normal forms for the group of basis-conjugating
automorphisms of a free group (with M. Gutierrez), submitted.
- 2.
- Cohomology of the string group (with B. Bogley), in
preparation.

*SUMMARY:*

In [1] we prove that the group *G*_{n} of automorphisms of *F*_{n} which take
every element of a fixed basis to a conjugate of itself has a regular
language of normal forms. (In another guise, *G*_{n} appears as the group of
motions of *n* circles in the 3-space and is of some importance in modern
physics [BL], [K].) This is a part of an ongoing research motivated by
the recent results showing that *Aut*(*F*_{n}) is not (bi)automatic
(Bridson-Vogtmann [BV]), while
its important subgroups, like the braid groups and the mapping class groups
of surfaces are automatic (Thurston [E+], Mosher [M]). It is still to be seen
if the
methods employed in [1] can be extended to provide an answer to the
questions of whether *G*_{n} is automatic, and whether *Aut*(*F*_{n}) has a regular
language of normal forms.

In [2], we prove that the cohomology ring of the above group *G*_{n} has a
simple presentation, as conjectured by Brownstein and Lee in [BL]: the
generators
and relators
and
.
The
proof uses the
description of abelian subgroups of *G*_{n} given in [1], the construction of
a constructible complex on which *G*_{n} acts with abelian cell stabilizers,
and the spectral sequence associated with the action. The proof also gives a
description of a free basis of every group *H*^{q}(*G*_{n}); it is in a bijective
correspondence with the set of rooted forests on *n* vertices which have
*n*-*q* components. A classical enumeration formula of Cayley implies then that
the Poincaré polynomial of *G*_{n} is
(1+*nt*)^{n-1}.

*REFERENCES:*

- [BV]
- M. Bridson and K. Vogtmann, On the geometry of the automorphism froup
of a free group, preprint 1994.
- [BL]
- A. Brownstein and R. Lee, Cohomology of the group of motions of
*n*strings in the 3-space, Contemp. Math. 150 (1993), 51-61. - [E+]
- D. B. A. Epstein et al., Word Processing in Groups, Bartlett and Jones,
Boston 1992.
- [L]
- H. Kauffman, Knots and Physics, World Scientific 1991.
- [M]
- L. Mosher, Mapping class groups are automatic, Annals of Math. 142 (1995), 303-384.